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Trigonometric form of complex numbers

Let $M (x,y)$ be the point in the complex plane which is joined to the complex number $z = x + yi$.

We can determine the position of the point $M$ (and thus of the complex number $z$) by using numbers $r$ and $\varphi$, where $r = |z|= \sqrt{x^2+y^2}$ (the distance from the point $M$ to the origin) and $\varphi \in \left [0, 2\pi \right \rangle $ is an angle between the segment line $\overline{OM}$ and positive part of the real axis. Number $r$ is called modulus of a complex number and angle $\varphi$ is called an argument of a complex number and it is denoted by $\varphi = arg(z)$.

 

Then

$$\cos \varphi = \frac{x}{r} \Rightarrow x = r \cos \varphi $$

and

$$\sin \varphi = \frac{y}{r} \Rightarrow y = r \sin \varphi$$

is valid.

Substituting in the expression $z=x + yi$, we obtain the trigonometric form of the complex number:

$$ z= r (\cos \varphi + i \sin \varphi).$$

If  a complex number is given in the algebraic form $ z = x+yi$, then we determine $r$ and $\varphi$ from equations:

$$r = \sqrt{x^2 + y^2}$$

$$tg \varphi = \frac{y}{x} , x \neq 0.$$

The last equation gives two solutions for an angle $\varphi \in \left [0, 2\pi \right \rangle$. We choose the angle depending on in which quadrant number $z$ is located.

Example 1: Write in the trigonometric form complex numbers $z$ and $\overline{z}$, if $z = \frac{1}{2} – \frac{\sqrt{3}}{2}i$.

Solution:

We need to determine numbers $r$ and $\varphi$.

$$r = |z| =\left|\frac{1}{2} – \frac{\sqrt{3}}{2}i\right| $$

$$= \sqrt{\left( \frac{1}{2} \right) ^2 + \left( -\frac{\sqrt{3}}{2} \right)^2 } $$

$$= \sqrt{\frac{1}{4}  + \frac{3}{4}} $$

$$= 1. $$

 

$$tg \varphi = \frac{-\frac{\sqrt{3}}{2}}{\frac{1}{2}} = -\sqrt{3},$$

that is $\varphi = \frac{2\pi}{3}$ or $\varphi = \frac{5\pi}{3}$. Since the number $z = \frac{1}{2} – \frac{\sqrt{3}}{2}i$ is located in the fourth quadrant, it follows that $\varphi = \frac{5\pi}{3}$.

The complex number $z= \frac{1}{2} – \frac{\sqrt{3}}{2}i$ has the trigonometric form:

$$ z = \cos \frac{5 \pi}{3} + i \sin \frac{5\pi}{3}.$$

The complex conjugate numbers have the same modulus. Therefore, for $\overline{z} = \frac{1}{2} + \frac{\sqrt{3}}{2}i$ , $r =1$. $\overline{z}$ is located in the first quadrant, so we have:

$$tg \varphi = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3} \Rightarrow \varphi = \frac{\pi}{3}.$$

Finally, the complex number $\overline{z} = \frac{1}{2} + \frac{\sqrt{3}}{2}$ has the following trigonometric form:

$$\overline{z} = \cos \frac{\pi}{3} + i \sin \frac{\pi}{3}.$$

Example 2: Write in the trigonometric form the complex number $z$:

$$ z = – \cos \frac{\pi}{5} + i \sin \frac{\pi}{5}.$$

Solution:

The function cosine is negative in the second and third quadrant, and sine is positive in the first and second quadrant. This means that the given complex number $z$ is located in the second quadrant.

$r = \sqrt{\left( -cos \frac{\pi}{5}\right) ^2 + \left( sin\frac{\pi}{5} \right)^2 } = \sqrt{1} = 1$

Now we have:

$$tg \varphi = \frac{\sin \frac{\pi}{5}}{-\cos \frac{\pi}{5}} = – tg \frac{\pi}{5}.$$

That is, $ \varphi = – \frac{\pi}{5}$ or $\varphi = \frac{4\pi}{5}$.

We know that the complex number $z$ is located in the second quadrant, which means that $ \varphi = \frac{4\pi}{5}$.

Now we can write the given complex number $z$ in the trigonometric form:

$$ z = \cos \frac{4 \pi}{5} + i \sin \frac{4 \pi}{3}.$$